Answer to Question #232311 in Math for abi

Question

L{t² cost t}

Answer 1

$$F(s)=L\{t^2\cos t\}=(-1)^2\dfrac{d^2}{ds^2}(L\{\cos t\})$$

$$=\dfrac{d^2}{ds^2}(\dfrac{s}{s^2+1})=\dfrac{d}{ds}(\dfrac{s^2+1-2s^2}{(s^2+1)^2})$$

$$=\dfrac{d}{ds}(\dfrac{1-s^2}{(s^2+1)^2})$$

$$=\dfrac{-2s(s^2+1)^2-4s(s^2+1)(1-s^2)}{(s^2+1)^4}$$

$$=\dfrac{-2s^3-2s-4s+4s^3}{(s^2+1)^3}$$

$$=\dfrac{2s^3-6s}{(s^2+1)^3}$$

$$L\{t^2\cos t\}=\dfrac{2s^3-6s}{(s^2+1)^3}=\dfrac{2s(s^2-3)}{(s^2+1)^3}$$